I have a Wii as well as next gen consoles, and continue to use my Wii occasionally. The YouTube channel on the Wii has always been dodgy. For instance, trying to fast forward or rewind a video has never worked for me. There is a thread on Google dating back over a year with people having issues with the Wii YouTube Channel, and while it got better, it was never really fixed completely. It doesn't help that Nintendo killed updates and online services for the Wii a mere 6-7 months after the Wii U was released, so it is, unlike the X360 and PS3, an abandoned platform.
That said, I just started my Wii and went on YouTube. I could see all of the channels I am subscribed to, but when I tried to watch videos, it would hang at the advert and never load the desired video. If there wasn't an ad, they would work after a 15 second delay or so.
The best I can do is offer a fairly lame suggestion: you can try to delete the channel, then reinstall it again from the Wii Shop channel. Make sure to go into settings and delete all the save data as well and not just the channel.